I am reading the survey of J. Ayoub, The motivic nearby cycles and the conservation conjecture (see here), in which he introduced the original version motivic nearby cycles (another note by Illusie is also a good introductory text) for $D^+(-,\Lambda)$ for $\Lambda$ a torsion ring. Let $S$ be the spectrum of a (strictly) henselian DVR (discrete valuation ring). We denote by $j:\eta \to S$ the generic point of $S$ and by $i:s \to S$ the closed point. We fix a separable closure $\overline{\eta}$ of $\eta$. Let $f:X \to S$ be a morphism of finite type. We form the diagram (by base change)
The nearby cycles functors $R\Psi_f:D^+(X_{\eta},\Lambda) \to D^+(X_s,\Lambda)$ is defined to be $\overline{i}^*R\overline{j}_*(A_{X_{\eta}})$. Ayoub then tried to define the nearby cycle for some stable homotopical $2$-functors, for instance,
- The motivic stable homotopy category $\mathbf{SH}(-)$ of Morel and Voevodsky;
- The stable category of mixed motives $\mathbf{DM}(-)$ of Voevodsky.
They are all unital monoidal triangulated categories. For rough definitions of these functors, one can consult his note or even his thesis on $\mathbf{SH}(-)$. There is at least one serious problem if one tries to mimic the definition in the case of $D^+(-,\Lambda)$, which is also a stable homotopical $2$-functor in the sense of Ayoub's thesis, is that the resulting functor may not be monoidal. More precisely, if one defines $$\Phi_f: \mathbf{DM}(X_{\eta}) \longrightarrow \mathbf{DM}(X_s)$$ simply by taking $\Phi_f(A) = \overline{i}^*\overline{j}_*A_{\mid X_{\eta}}$ then $\Phi_f(\mathbb{Z}) \neq \mathbb{Z}$ where $\mathbb{Z}$ denotes the unit of $\mathbf{DM}$ (the image of the base scheme). Let's take $S$ to be the henselization of the affine line over $\mathbb{A}^1_k = \mathrm{Spec}(k[T])$ ($k=\overline{k}$ and $\mathrm{Char}(k)=0$). Ayoub noted that this is an easy computation, one sees that in this case $\overline{S}$ is the limit of $S[T^{1/n}]$ (this is where the hypothesis of being henselian DVR applied), one forms the diagram
where $e_n$ denotes the $n$-th power map. Then $\Psi_{\mathrm{id}}\mathbb{Z}$ is the colimit of $i^*_n j_{n*}\mathbb{Z}$. Here it comes to the point I do not understand: why $i^*_n j_{n*}\mathbb{Z} = \mathbb{Z} \oplus \mathbb{Z}(-1)[-1]$ ($(-1)$ means the Tate twist, it makes sense for any stable homotopical $2$-functor via Thom equivalence, and $[-1]$ means the inverse functor of the translation in a triangulated category) and for $n \mid m$, the canonical morphism $i^*_n j_{n*}\mathbb{Z} \to i^*_n j_{n*}\mathbb{Z}$ is given by the matrix
$$\begin{pmatrix} 1 & 0 \\ 0 & m/n \\ \end{pmatrix}: \mathbb{Z} \oplus \mathbb{Z}(-1)[-1] \longrightarrow \mathbb{Z} \oplus \mathbb{Z}(-1)[-1]$$
and hence $\Phi_{\mathrm{id}}\mathbb{Z} = \mathbb{Z} \oplus \mathbb{Q}(-1)[-1]$. This situation does not appear in etale context because of torsion hypothesis. So I would like to ask whether someone can help me on this computation. Any help is appreciate. Thank you in advance.
